RECENT ADVANCES IN SCIENCE 117 



Company, 12s. net) was published in 1914. It is particularly 

 interesting to read of the approaches made by Japanese mathe- 

 matics to the infinitesimal calculus, of which an embryonic form 

 was known as "yenri"; and it may also be noticed that deter- 

 minants were invented in Japan before Leibniz in Europe had 

 any idea of them. In the Monist for 1914 and 1915 there have 

 been several articles in which the development of the principles 

 of mechanics with Newton and his contemporaries is treated in 

 great detail, and the papers in question will be shortly collected 

 and republished. The most interesting feature is an extract, 

 in the number for October 1914, from a manuscript of Newton's 

 which has been hitherto unpublished, showing that the im- 

 portant conception of " mass " developed much earlier with 

 Newton than is generally supposed. The whole question of the 

 principles of mechanics with Newton is extremely closely con- 

 nected with Newton's mathematical work ; and Newton's mathe- 

 matical work is also considered in some detail in the annotated 

 Essays on the Life and Work of Newton by Augustus De Morgan 

 (Chicago and London, the Open Court Publishing Company, 

 55. net), which has been already reviewed in this magazine 

 (April 191 5, pp. 687-9). A very valuable monograph on 

 Aristarchus of Santos : the Ancient Copernicus, by Sir Thomas 

 Heath, was published in 191 3 by the Clarendon Press. 



Vol. v. (191 3) of the Journal of the Indian Mathematical Society 

 contains a good address by Prof. G. A. Miller, given to the 

 Illinois Society, and entitled " Some Thoughts on Modern 

 Mathematical Research"; vol. vi. (1914) contains the first part 

 of an historical and critical study of the various theories of 

 irrational number by Philip E. B. Jourdain. In the volumes 

 of this Journal are a great number of short notes, and there 

 are not very many longer articles. In the current volume (vii.) 

 there is a long paper on " So-called Cases of Failure in the 

 Solution of Linear Differential Equations " by Eric H. Neville. 



We will now consider more particularly some work on 

 algebra and the theory of numbers. One of the mathematical 

 papers in Section A of vol. xxxii. (1914) of the Proceedings of the 

 Royal Irish Academy is by M. W. J. Fry, and bearing the title 

 " Real and Complex Numbers Considered as Adjectives or 

 Operators," carries our thoughts back to the first half of last 

 century. The object of this paper is to define the symbols 

 + , — , and i in such a way "that the rules to be followed in 



